Step 1: Set up the diffusivity equation and treat dp/dt as a given constant $C$:
Start from \( \frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial p}{\partial r}\right) = \frac{\phi \mu c_t}{k}\frac{\partial p}{\partial t} \). If $\frac{\partial p}{\partial t} = C$ is treated as a constant that does not depend on $r$ or $t$, the right hand side becomes a fixed number, and the equation reduces to an ordinary differential equation in $r$ alone, $\frac{1}{r}\frac{d}{dr}\left(r\frac{dp}{dr}\right) = \frac{\phi \mu c_t}{k} C$.
Step 2: Consider the special case $C = 0$:
If the constant $C$ is zero, then $\frac{\partial p}{\partial t} = 0$ everywhere, meaning pressure at every radius no longer changes with time. This is exactly the definition of the steady state, which happens physically when the reservoir has a constant pressure outer boundary, such as a strong active aquifer or a constant pressure injection boundary, that keeps replenishing the produced fluid. So the steady state case, option (C), is a valid solution consistent with dp/dt being constant.
Step 3: Consider the case $C \ne 0$ but still constant:
If $C$ is a fixed nonzero number, pressure is declining at the same rate everywhere in the reservoir at any instant in time, though it still varies with radius. This is exactly the pseudo-steady state, which arises physically once the pressure transient has reached a closed, no flow outer boundary and the whole reservoir volume depletes uniformly with time. So the pseudo-steady case, option (B), is also consistent with dp/dt being constant.
Step 4: Rule out the transient and unsteady options:
In transient or unsteady flow, the pressure disturbance is still propagating outward and has not stabilized, so $\frac{\partial p}{\partial t}$ genuinely depends on both $r$ and $t$ and cannot be pulled out as a single constant. This rules out options (A) and (D).
Step 5: Conclude:
Only the pseudo-steady state (constant nonzero decline rate) and the steady state (constant zero decline rate, i.e. no change) are consistent with treating dp/dt as constant.
Final Answer:
\[ \boxed{\text{(B) Pseudo-steady and (C) Steady}} \]